We study the minimal number of variables required by a totally positive definite diagonal universal quadratic form over a real quadratic field $\Bbb Q(\sqrt{D})$ and obtain lower and upper bounds for it in terms of certain sums of coefficients of the associated continued fraction. We also estimate such sums in terms of $D$ and establish a link between continued fraction expansions and special values of $L$-functions in the spirit of Kronecker's limit formula.

11E12, 11R11, 11A55

universal quadratic form, real quadratic form, number field, continued fraction, additively indecomposable integer, Kronecker's limit formula

• 1. M. Bhargava, On the Conway-Schneeberger Fifteen Theorem, Contemp. Math. 272 (1999), 27-37. zbl 0987.11027; MR 1803359.
• 2. M. Bhargava, J. Hanke, Universal quadratic forms and the 290-theorem, preprint.
• 3. A. Biró, A. Granville, Zeta functions for ideal classes in real quadratic fields, at $s=0$, J. Number Theory 132 (2012), 1807-1829. DOI 10.1016/j.jnt.2012.02.003; zbl 1276.11180; MR 2922348.
• 4. V. Blomer, G. Harcos, P. Michel, Bounds for modular $L$-functions in the level aspect, Ann. Sci. Ecole Norm. Sup. 40 (2007), 697-740. DOI 10.1016/j.ansens.2007.05.003; zbl 1185.11034; MR 2382859.
• 5. V. Blomer, V. Kala, Number fields without universal $n$-ary quadratic forms, Math. Proc. Cambridge Philos. Soc. 159 (2015), 239-252. DOI 10.1017/S030500411500033X; zbl 1371.11084; MR 3395370; arxiv 1503.05736.
• 6. W. K. Chan, M.-H. Kim, S. Raghavan, Ternary universal integral quadratic forms, Japan. J. Math. 22 (1996), 263-273. MR 1432376.
• 7. A. Dress, R. Scharlau, Indecomposable totally positive numbers in real quadratic orders, J. Number Theory 14 (1982), 292-306. DOI 10.1016/0022-314X(82)90064-6; zbl 0507.12002; MR 0660374.
• 8. W. Duke, J. Friedlander, H. Iwaniec, The subconvexity problem for Artin $L$-functions, Invent. Math. 149 (2002), 489-577. DOI 10.1007/s002220200223; zbl 1056.11072; MR 1923476.
• 9. C. Friesen, On continued fractions of given period, Proc. Amer. Math. Soc. 103 (1988), 8-14. DOI 10.1090/S0002-9939-1988-0938635-4; zbl 0652.10006; MR 0938635.
• 10. E. Fogels, On the zeros of Hecke's L-functions I, Acta Arith. 7 (1962), 87-106. zbl 0100.03801; MR 0136585.
• 11. G. H. Hardy, E. M. Wright, An introduction to the theory of numbers, 5th edition. The Clarendon Press, Oxford University Press, New York, 1979.
• 12. Heath-Brown, Hybrid bounds for Dirichlet $L$-functions. II, Quart. J. Math. Oxford 31 (1980), 157-167. zbl 0396.10030; MR 0576334.
• 13. E. Hecke, Über die Kroneckersche Grenzformel für reelle quadratische Körper und die Klassenzahl relativ-abelscher Körper, Verhandl. d. Naturforschenden Gesell. Basel 28, 363-372 (1917). (Mathematische Werke pp.198-207). zbl 47.0144.01.
• 14. G. Herglotz, Über die Kroneckersche Grenzformel für reelle, quadratische Körper I, II, Berichte Verhandl. Sächs. Akad. Wiss. Leipzig 75 (1923), 3-14, 31-37. zbl 49.0125.03.
• 15. S. W. Jang, B. M. Kim, A refinement of the Dress-Scharlau theorem, J. Number Theory 158 (2016), 234-243. DOI 10.1016/j.jnt.2015.06.003; zbl 1331.11092; MR 3393549.
• 16. M. Jacobson, H. Williams, Solving the Pell equation, CMS books in mathematics, Springer-Verlag 2009. ISBN 978-0-387-84922-5. MR 2466979.
• 17. V. Kala, Universal quadratic forms and elements of small norm in real quadratic fields, Bull. Aust. Math. Soc. 94 (2016), 7-14. DOI 10.1017/S0004972715001495; zbl 1345.11025; MR 3539315.
• 18. V. Kala, Norms of indecomposable integers in real quadratic fields, J. Number Theory 166 (2016), 193-207. DOI 10.1016/j.jnt.2016.02.022; zbl 06572137; MR 3486273; arxiv 1512.04691.
• 19. B. M. Kim, Finiteness of real quadratic fields which admit positive integral diagonal septenary universal forms, Manuscr. Math. 99 (1999), 181-184. DOI 10.1007/s002290050168; zbl 0961.11016; MR 1697212.
• 20. B. M. Kim, Universal octonary diagonal forms over some real quadratic fields, Commentarii Math. Helv. 75 (2000), 410-414. DOI 10.1007/s000140050133; zbl 1120.11301; MR 1793795.
• 21. X. Li, Upper bounds on $L$-functions at the edge of the critical strip, IMRN 2010, 727-755. zbl 1219.11136; MR 2595006.
• 22. J. E. Littlewood, On the class-number of the corpus $P(\sqrt{-k})$, Proc. Lond. Math. Soc 27 (1928), 358-372. DOI 10.1112/plms/s2-27.1.358; zbl 54.0206.02; MR 1575396.
• 23. H. Maaß, Über die Darstellung total positiver Zahlen des Körpers $R(\sqrt 5)$ als Summe von drei Quadraten, Abh. Math. Sem. Univ. Hamburg 14 (1942), 185-191. DOI 10.1007/BF02940744; zbl 67.0103.02; MR 3069720.
• 24. O. Perron, Die Lehre von den Kettenbrüchen, Chelsea, New York, 1958.
• 25. H. Sasaki, Quaternary universal forms over $\mathbb Q[\sqrt{13}]$, Ramanujan J. 18 (2009), 73-80. DOI 10.1007/s11139-007-9056-2; zbl 1193.11033; MR 2471617.
• 26. C. L. Siegel, Sums of $m$-th powers of algebraic integers, Ann. Math. 46 (1945), 313-339. zbl 0063.07010; MR 0012630.
• 27. P. Yatsyna, A lower bound for the rank of a universal quadratic form with integer coefficients in a totally real field, preprint.
• 28. D. Zagier, A Kronecker limit formula for real quadratic fields, Math. Ann. 213 (1975), 153-184. DOI 10.1007/BF01343950; zbl 0283.12004; MR 0366877.
• 29. D. Zagier, Zetafunktionen und quadratische Körper, Springer-Verlag 1981, ISBN 3-540-10603-0 MR 0631688.

University of Göttingen, Mathematisches Institut, Bunsenstr. 3-5, D-37073 Göttingen, Germany

University of Göttingen, Mathematisches Institut, Bunsenstr. 3-5, D-37073 Göttingen, Germany and Charles University, Faculty of Mathematics and Physics, Department of Algebra, Sokolovská 83, 18600 Praha 8, Czech Republic